Chapter 3: Position, Speed and
Velocity
 3.1 Space and Position
 3.2 Graphs of Speed and Velocity
 3.3 Working with Equations
Chapter Objectives

Calculate time, distance, or speed when given two of the three values.

Solve an equation for any of its variables.

Use and interpret positive and negative values for velocity and position.

Describe the relationship between three-dimensional and onedimensional systems.

Draw and interpret graphs of experimental data, including velocity
versus position, and speed versus time.

Use a graphical model to make predictions that can be tested by
experiments.

Derive an algebraic model from a graphical model and vice versa.

Determine velocity from the slope of a position versus time graph.

Determine distance from the area under a velocity versus time graph.
Chapter Vocabulary
 average speed
 origin
 constant speed
 position
 coordinates
 rate
 coordinate system
 slope
 displacement
 time
 instantaneous speed
 vector
 instantaneous velocity
 velocity
Inv 3.1 Position, Speed, and Velocity
Investigation Key Question:
How are position, speed, and velocity related?
3.1 Space and position
 In physics, the word position refers to the
location of an object at one instant.
 A position is always specified relative to an
origin.
 The net change in position relative to the origin
is called displacement.
3.1 Position and distance
 Distance is related to, but different from, position.
 Distance is a measure of length without regard to
direction.
3.1 Position in three dimensions
 Space is three
dimensional, so position
must also be a threedimensional variable.
 Any position in space
can be precisely
specified with three
numbers called
coordinates.
3.1 Positive and negative
 Allowing x, y, and z to have positive and
negative values allows coordinates to locate any
position in all of space.
3.1 One dimensional problems
 In three-dimensional space, position is a vector.
 A vector is a variable that contains all three
coordinate values.
 Motion in a straight line is easiest to analyze
because it is one dimensional.
 However, even in one dimension there is an
origin and positive and negative values are
possible.
3.1 Speed and distance
 Speed is the rate at which distance changes.
 In physics, the word rate means the ratio of how
much something changes divided by how long
the change takes.
 Constant speed means the same change in
distance is traveled every second.
3.1 Calculating speed
 The change in position is a
distance traveled in a given
amount of time.
 To calculate the speed of an
object, you need to know
two things:
 the distance traveled by
the object
 the time it took to travel
the distance
3.1 Calculating speed
 Since speed is a ratio of distance over time, the
units for speed are a ratio of distance units over
time units.
Calculating speed in meters
per second

A bird is observed to fly 50 meters in 7.5 seconds.
Calculate the speed of the bird in m/sec.
1.
You are asked for speed in m/s.
2.
You are given distance = 50 m; time = 7.5 s
3.
Use v = d ÷ t
4.
Plug in values and solve. v = 50 m ÷ 7.5 s ≈ 6.67 m/s
3.1 The velocity vector
 The velocity of an object
tells you both its speed and
its direction of motion.
 A velocity can be positive or
negative.
 The positive or negative sign
for velocity is based on the
calculation of a change in
position.
Two cars going opposite
directions have the same
speed, but their
velocities are different—
one is positive and the
other is negative.
3.1 The velocity vector
 Velocity is the change in position divided by the
change in time.
Chapter 3: Position, Speed and
Velocity
 3.1 Space and Position
 3.2 Graphs of Speed and Velocity
 3.3 Working with Equations
Inv 3.2 Position, Velocity, and Time
Graphs
Investigation Key Question:
How are graphs used to describe motion?
3.2 Graphs of Speed and Velocity
 There are many graphs involving the terms
speed, velocity, distance, position, displacement
and time.
 A position versus time graph shows the details
of the actual motion during the trip.
3.2 Average vs. instantaneous speed
 Average speed is the
total distance traveled
divided by the total time
taken.
 Instantaneous speed is
the apparent speed at
any moment, such as on
a speedometer.
Interpreting a distance versus
time graph
This distance versus time graph shows a boat traveling
through a long canal. The boat has to stop at locks for
changes in water level.
1.
How many stops does it make?
2.
What is the boat’s average
speed for the whole trip?
3.
What is the highest speed the
boat reaches?
Interpreting a distance versus
time graph
1. The boat makes three stops because there are
three horizontal sections on the graph.
2. The average speed is 10 km/h (100 km ÷ 10 h).
3. The highest speed is 20 km/h. The position
changes by 20 km in one hour for the first, third,
and fifth hours of the trip.
3.2 Slope
 The slope of a line is
the ratio of the “rise”
(vertical change) to
the “run”(horizontal
change) of the line.
3.2 Speed is the slope of the distance
versus time graph
3.2 Positive and negative velocities
 When the direction of motion is part of the
calculation, changes in position are referred to
as displacement.
3.2 Positive and negative velocities
 Average velocity uses the values of
displacement and elapsed time from the position
vs. time graph.
 The average velocity at C is 12 mph.
3.2 Positive and negative velocities
 The slope of the position vs. time graph at any
one time is called instantaneous velocity.
3.2 Velocity Equations
 Velocity (v) is calculated
by dividing the change in
position (Δx) by the
change in time (Δt).
3.2 The velocity versus time graph
 The velocity versus time graph has velocity on
the y-axis and time on the x-axis.
 On this graph, a constant velocity is a straight
horizontal line.
 Information about an object’s position is also
present in the velocity versus time graph.
3.3 Constant Velocity

This graph shows that
the velocity:
1. is 1 m/s.
2. stays constant at 1
m/s for 10 seconds.
3.2 The velocity vs. time graph
 The area on a velocity versus time graph is
equal to the distance traveled.
3.2 Relating v vs. t
 A velocity versus time
graph can show
positive and negative
velocities.
3.2 Relating x vs. t
 The position versus
time graph, can
yield the same
information using
the slope to
calculate velocity at
corresponding time
intervals.
Chapter 3: Position, Speed and
Velocity
 3.1 Space and Position
 3.2 Graphs of Speed and Velocity
 3.3 Working with Equations
Inv 3.3 Equations of Motion
Investigation Key Question:
How are equations used in physics?
3.3 Working with Equations
 An equation is a much more powerful form of
model than a graph.
 While graphs are limited to two variables,
equations can have many variables and can
be used over a wide range of values.
3.3 Working with Equations
 Equations can also be rearranged to show
how any one variable depends on all the
others.
Calculating time from speed
and distance
 How far do you go if you drive for 2 h at a speed of
100 km/h?
1.
You are asked for distance.
2.
You are given time in h and speed in km/h.
3.
Use d = vt.
4.
Solve. d = 2 h × 100 km/h = 200 km
3.3 Solving an equation
 To “solve” means to get a desired variable by
itself on one side of an equals sign.
 Whatever you do to the left of the equals sign
you must do exactly the same to the right.
 Get in the habit of solving an equation before
you plug in numbers.
 More complex problems require you to
substitute whole equations for single variables.
3.3 Solving an equation
 To solve this equation for
distance (d):
 Multiply both sides of the
equation by “t”.
 Multiplying by “t”on both
sides of the equation allows
you to cancel a t from the
numerator and the
denominator on the right
side of the equation.
3.3 Position vs. time equation
 The equation says your position, x, is equal to
the position you started at, x0, plus the
additional amount you traveled, vt.
Calculating time from speed
and distance

A car moving in a straight line at constant
velocity starts at a position of 10 meters
and finishes at 30 meters in five seconds.
What is the velocity of the car?
1.
You are asked for velocity.
2.
You are given that the motion is at constant velocity, two positions,
and the time.
3.
Use x = x0 + vt, solve for v.
4.

x – x0 = vt

x – x0 = v
t
Substitute numbers for variables: v = 30 m – 10 m = 4 m/s
5s
3.3 Relating equations and graphs
 In science and engineering, any two variables
can be used in the equation for a line, not just x
and y.
3.3 Relating equations and graphs
 The y corresponds to x, the position at any time;
 the x corresponds to time “t” ;
 the slope, m, corresponds to the velocity, v;
 the y-intercept, b, corresponds to the initial position, x0.
3.3 Scientific process
 The process of
developing a model or
theory in science
starts with actual
experiments and data,
and produces a
validated model in the
form of an equation.
3.3 How to solve physics problems
 Step 1
 Identify clearly what the problem is asking.
 Step 2
 Identify the information you are given.
 Step 3
 Identify relationships.
 Step 4
 Combine the given information and the
relationships.
Calculating distance from time
and speed

A space shuttle is traveling at a speed of 7,700 meters per
second.

How far in kilometers does the shuttle travel in one hour?

At an altitude of 300 kilometers, the circumference of the
shuttle’s orbit is 42 million meters.

How long does it take the shuttle to go around Earth one
time?
Calculating distance from time
and speed
1.
This is a two-part problem asking for distance in kilometers and time
in hours.
2.
You are given a speed and time for the first part, and a speed and
distance for the second.
3.
d = vt, and t = d ÷ v
1 h = 3,600 s
1 km = 1,000 m
4.
Part 1:
 d = (7,700 m/s)(3,600 s) = 27,720,000 m
 Convert to kilometers: 27,720,000 m ÷ 1,000 km/m = 27,720 km
5.
Part 2:
 t = 42 × 106 m ÷ 7,700 m/s = 5,455 s
 Convert to minutes: = 5,455s ÷ 60 s/min = 90.9 minutes
Slow Motion Photography
 A video camera does not photograph moving images.
 It takes a sequence of still images called frames and
changes them fast enough that your brain perceives a
moving image.
 You can use an ordinary video camera to analyze
motion in laboratory experiments.